A264990 a(n) = number of occurrences of a most frequent nonzero digit in factorial base representation (A007623) of n.

0, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Author

Antti Karttunen, Dec 22 2015

Keywords

• nonn
• base

Examples

			   n  A007623(n)   a(n) [highest number of times any nonzero digit occurs].
   0 =   0           0 (because no nonzero digits present)
   1 =   1           1
   2 =  10           1
   3 =  11           2
   4 =  20           1
   5 =  21           1
   6 = 100           1
   7 = 101           2
   8 = 110           2
   9 = 111           3
  10 = 120           1
  11 = 121           2
  12 = 200           1
  13 = 201           1
  14 = 210           1
  15 = 211           2
  16 = 220           2
  17 = 221           2
  18 = 300           1
and for n=63 we have:
  63 = 2211          2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10080
• Index entries for sequences related to factorial base representation.

Crossrefs

Cf. A007623, A051903, A225901, A257511, A257684, A275735, A275811.

Cf. A265349 (positions of terms <= 1), A265350 (positions of term > 1).

Cf. also A266117, A266118.

Programs

Mathematica

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Max[Tally[Select[s, # > 0 &]][[;;,2]]]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jan 24 2024 *)

Python

from sympy import prime, factorint
from operator import mul
import collections
def a007623(n, p=2): return n if n
Indranil Ghosh, Jun 20 2017

Formula

a(0) = 0; for n >= 1, a(n) = max(A257511(n), a(A257684(n))).

Other identities. For all n >= 0:

From Antti Karttunen, Aug 15 2016: (Start)

a(n) = A275811(A225901(n)).

a(n) = A051903(A275735(n)).

(End)

Extensions

Name changed by Antti Karttunen, Aug 15 2016